协议的复杂性
The Complexity of Agreement
摘要：
1976年，奥曼（Aumann）提出了一个著名的定理，即具有共同先验的诚实、理性的贝叶斯代理人永远不会“同意不同意”：如果他们对任何主题的意见是常识，那么这些意见必须是平等的。经济学家已经写了许多论文来研究这个定理背后的假设。但有两个关键问题没有得到解决：第一，代理人能否在合理长度的对话后达成协议？第二，对话所需的计算是否能够有效地执行？本文对这两个问题作了肯定的回答，从而加强了奥曼的原始结论。我们首先证明，对于两个具有共同先验的智能体，如果它们对一个[0，1]变量的期望值的一致性在1/2以内，那么它们交换1/ε ^2阶比特就足够了。这个界限是完全独立的相关知识的位数n的代理。然后，我们将约束扩展到三个或更多的代理人;我们给出了一个例子，经济学家的“标准协议”（包括反复宣布一个人的当前预期）几乎饱和的约束，而一个新的“衰减协议”做得更好。最后，我们给出了一个协议，该协议在交换1/ε ^2阶的消息后，会使两个贝叶斯人在最小范围内达成一致，并且该协议可以由计算资源有限的代理来模拟。我们的意思是，在检查了代理人的知识和他们的谈话记录之后，没有人能够区分代理人和完美的贝叶斯人。模拟过程所用的时间与1/ε ^[6]成指数关系，但与n无关。
英文摘要：
A celebrated 1976 theorem of Aumann asserts that honest, rational Bayesian agents with common priors will never "agree to disagree": if their opinions about any topic are common knowledge, then those opinions must be equal. Economists have written numerous papers examining the assumptions behind this theorem. But two key questions went unaddressed: first, can the agents reach agreement after a conversation of reasonable length? Second, can the computations needed for that conversation be performed efficiently? This paper answers both questions in the affirmative, thereby strengthening Aumann's original conclusion. We first show that, for two agents with a common prior to agree within epsilon about the expectation of a [0,1] variable with high probability over their prior, it suffices for them to exchange order 1/epsilon^2 bits. This bound is completely independent of the number of bits n of relevant knowledge that the agents have. We then extend the bound to three or more agents; and we give an example where the economists' "standard protocol" (which consists of repeatedly announcing one's current expectation) nearly saturates the bound, while a new "attenuated protocol" does better. Finally, we give a protocol that would cause two Bayesians to agree within epsilon after exchanging order 1/epsilon^2 messages, and that can be simulated by agents with limited computational resources. By this we mean that, after examining the agents' knowledge and a transcript of their conversation, no one would be able to distinguish the agents from perfect Bayesians. The time used by the simulation procedure is exponential in 1/epsilon^6 but not in n.
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